Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

: Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, ﬁrst we deﬁne and study statistical solitons on Ricci-symmetric statistical warped products R × f N 2 and N 1 × f R . Second, we study statistical warped products as submanifolds of statistical manifolds. For statistical warped products statistically immersed in a statistical manifold of constant curvature, we prove Chen’s inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi–Civita connection). At the end, we establish a relationship between the scalar curvature and the Casorati curvatures in terms of the Laplacian of the warping function for statistical warped product submanifolds in the same ambient space. Writing-Review and Editing, A.N.S. and B.-Y.C.; Visualization, A.N.S.; Supervision, B.-Y.C.; Project Administration, A.N.S., B.-Y.C. and O.B.; Funding Acquisition, A.N.S., B.-Y.C. and O.B.


Introduction
Statistical manifolds were introduced in 1985 by S. Amari [1] in terms of information geometry, and they were applied by Lauritzen in [2]. Such manifolds have an important role in statistics as the statistical model often forms a geometrical manifold.
One of the most fruitful generalizations of Riemannian products is the warped product defined in [3]. The notion of warped products plays very important roles in differential geometry and in mathematical physics, especially in general relativity. For instance, space-time models in general relativity are usually expressed in terms of warped products (cf., e.g., [4,5]).
In 2006, L. Todjihounde [6] defined a suitable dualistic structure on warped product manifolds. Furthermore, Furuhata et al. [7] defined Kenmotsu statistical manifolds and studied how to construct such structures on the warped product of a holomorphic statistical manifold [8] and a line. In [9], H. Aytimur and C. Ozgur studied Einstein statistical warped product manifolds. Further, C. Murathan and B. Sahin [10] studied and obtained the Wintgen-like inequality for statistical submanifolds of statistical warped product manifolds.
The Ricci solitons are special solutions of the Ricci flow of the Hamilton. In Section 4, we define statistical solitons and study the problem under what conditions the base manifold or fiber manifold of a statistical warped product manifold is a statistical soliton.
Curvature invariants play the most fundamental and natural roles in Riemannian geometry. A fundamental problem in the theory of Riemannian submanifolds is (cf. [11]): Problem A."Establish simple optimal relationships between the main intrinsic invariants and the main extrinsic invariants of a submanifold." The first solutions of this problem for warped product submanifolds were given in [11,12]. In Section 5, we study this fundamental problem for statistical warped product submanifolds in any statistical manifolds of constant curvature. Our solution to this problem given in this section is derived via the fundamental equations of statistical submanifolds.
An extrinsic curvature of a Riemannian submanifold was defined by Casorati in [13], as the normalized square of the length of the second fundamental form. Casorati curvature has nice applications in computer vision. It was preferred by Casorati over the traditional curvature since it corresponds better to the common intuition of curvature.
Inspired by historical development on the classifications of Casorati curvatures and Ricci curvatures, we establish in Section 6 an inequality for statistical warped product submanifolds in a statistical manifold of constant curvature. In the last section, we provide two examples of statistical warped product submanifolds in the same environment.

Preliminaries
Let (Ñ,∇,g) be a statistical manifold and N be a submanifold ofÑ. Then, (N, ∇, g) is also a statistical manifold with the statistical structure (∇, g) on N induced from (∇,g), and we call (N, ∇, g) a statistical submanifold.
The fundamental equations in the geometry of Riemannian submanifolds are the Gauss and Weingarten formulae and the equations of Gauss, Codazzi, and Ricci (cf. [4,5,27]). In the statistical setting, the Gauss and Weingarten formulae are defined respectively by [28]: for any E, F ∈ Γ(TN) and ξ ∈ Γ(T ⊥ N), where∇ and∇ * (resp., ∇ and ∇ * ) are the dual connections onÑ (resp., on N).
The symmetric and bilinear imbedding curvature tensor of N inÑ with respect to∇ and∇ * is denoted as h and h * , respectively. The relation between h (resp. h * ) and A ξ (resp. A * ξ ) is defined by [28]:g for any E, F ∈ Γ(TN) and ξ ∈ Γ(T ⊥ N).
LetR and R be the curvature tensor fields of∇ and ∇, respectively. The corresponding Gauss, Codazzi, and Ricci equations are given by [28]: for any E, F, G, H ∈ Γ(TN) and ξ, η ∈ Γ(T ⊥ N), where R ⊥ is the Riemannian curvature tensor on T ⊥ N. Similarly,R * and R * are respectively the curvature tensor fields with respect to∇ * and ∇ * . We can obtain the duals of all Equations (3)-(5) with respect to∇ * and ∇ * . Furthermore, are respectively the curvature tensor fields ofÑ and N given by [7]. Thus, the sectional curvature K ∇,∇ * on N ofÑ is defined by [29,30]: for any orthonormal vectors E, F ∈ T p N, p ∈ N. Suppose that dim(N) = m and dim(Ñ) = n. Let {e 1 , . . . , e m } and {e m+1 , . . . , e n } be respectively the orthonormal basis of T p N and T ⊥ p N for p ∈ N. Then, the scalar curvature σ ∇,∇ * of N is given by: The normalized scalar curvature ρ of N is defined as: The mean curvature vectors H and H * of N inÑ are: Furthermore, we set: for i, j ∈ {1, . . . , m}, a ∈ {m + 1, . . . , n}.
A statistical manifold (Ñ,∇,g) is said to be of constant curvaturec ∈ R, denoted byÑ(c), if the following curvature equation holds:

Basics on Statistical Warped Product Manifolds
Definition 1. [3] Let (N 1 , g 1 ) and (N 2 , g 2 ) be two (pseudo)-Riemannian manifolds and f > 0 be a differentiable function on N 1 . Consider the natural projections π : N 1 × N 2 → N 1 and π : N 1 × N 2 → N 2 . Then, the warped product N = N 1 × f N 2 with warping function f is the product manifold N 1 × N 2 equipped with the Riemannian structure such that: for E, F ∈ Γ(T (u,v) N), u ∈ N 1 , and v ∈ N 2 , where * denotes the tangent map.
Let χ(N 1 ) and χ(N 2 ) be the set of all vector fields on N 1 × N 2 , which is the horizontal lift of a vector field on N 1 and the vertical lift of a vector field on N 2 , respectively. We have Thus, it can be seen that π * (χ(N 1 )) = Γ(TN 1 ) and Recall the following general result from [6] for a dualistic structure on the warped product manifold N 1 × f N 2 . Proposition 1. Let (g 1 , ∇ N 1 , ∇ N 1 * ) and (g 2 , ∇ N 2 , ∇ N 2 * ) be dualistic structures on N 1 and N 2 , respectively. For X, Y ∈ χ(N 1 ) and U, V ∈ χ(N 2 ), D, D * on N 1 × N 2 satisfy: Furthermore, Todjihounde [6] derived the curvature of the statistical warped productÑ = N 1 × f N 2 in terms of the curvature tensors R 1 and R 2 of N 1 and N 2 , respectively, and its warping function f.
whereR denotes the curvature tensor field of (Ñ = N 1 × f N 2 , D, D * ,g) and The next result from [9] provides the Ricci tensorRic of the statistical warped product manifold.
where Ric 1 and Ric 2 are the Ricci tensors of N 1 and N 2 , respectively, and ∆f = div(grad f) is the Laplacian of f with respect to D.
We recall the following result from [31]. This result is useful in some Riemannian problems like the study of the distance between two manifolds, of the extremes of sectional curvature and is applied successfully in the demonstration of the Chen inequality.
Let (N, g) be a Riemannian submanifold of a Riemannian manifold (Ñ,g), and let f :Ñ → R be a differentiable function. Let: be the constrained extremum problem.
Theorem 1. If x ∈ N is the solution of the problem (11), then: is positive semi-definite, where h is the second fundamental form of N inÑ and grad f denotes the gradient of f .

Statistical Solitons on Statistical Warped Product Manifolds
The Ricci solitons model the formation of singularities in the Ricci flow, and they correspond to self-similar solutions. R. Hamilton [32] introduced the study of Ricci solitons as fixed or stationary points of the Ricci flow in the space of the metrics on Riemannian manifolds modulo diffeomorphisms and scaling. Since then, many researchers studied Ricci solitons for different reasons and in different ambient spaces (for example [33][34][35]). A complete Riemannian manifold (Ñ,g) is called a Ricci soliton (Ñ,g, ζ, λ) if there exists a smooth vector field ζ and a constant λ ∈ R such that: where L ζ denotes the Lie derivative along ζ andRic is the Ricci tensor ofg.
Based on these, we have the following.
The main purpose of this section is to study the problem: under what conditions does the base manifold or fiber manifold of the statistical warped product manifold become a statistical soliton?
Denote the Ricci-symmetric statistical warped product manifold by given by (12) and (13).
It follows from Lemma 2 that the Ricci tensor ofÑ is given as below: Thus, (12) and (13) can be rewritten as: and: respectively. Throughout this section, we use the statistical warped products as Ricci-symmetric. We give the following results by applying Lemma 2: is a trivial statistical manifold of dimension one and dim(N 2 ) = k. Then, for U, V ∈ χ(N 2 ), we have: Proposition 2. Let (ζ, λ) be a statistical soliton on statistical warped product manifold Then: Proof. SinceÑ is a statistical soliton, then from (6), we have: By taking into account Lemma 3 and Ric 1 (∂z, ∂z) = 0, we get: which gives Hess f (∂z, ∂z) = ( fλ k )g(∂z, ∂z).
An immediate consequence of Theorem 3 is as follows: is a statistical soliton.

B.Y. Chen Inequality
A universal sharp inequality for submanifolds in a Riemannian manifold of constant sectional curvature was established in [38], known as the first Chen inequality. The main purpose of this section is to establish the corresponding inequality for statistical warped product manifolds statistically immersed in a statistical manifold of constant curvature.
Let ϕ : N = N 1 × f N 2 →Ñ(c) be an isometric statistical immersion of a warped product N 1 × f N 2 into a statistical manifold of constant sectional curvaturec. We denote by r, k, and m = r + k the dimensions of N 1 , N 2 , and N 1 × N 2 , respectively. Since N 1 × f N 2 is a statistical warped product, we have: for unit vector fields E 1 and E 2 tangent to N 1 and N 2 , respectively. Hence, we derive: If we choose a local orthonormal frame {e 1 , . . . , e m } such that {e 1 , . . . , e r } are tangent to N 1 and {e r+1 , . . . , e r+k = e m } are tangent to N 2 , then we have: for each j = r + 1, . . . , m.
We rewrite the terms of the RHS of the previous equation as: Since, 2h 0 = h + h * , we get: Thus, we have: Using the Gauss equation for the Levi-Civita connection, we arrive at: which can be rewritten as: Substituting (23) into (22), we get: Furthermore, we derive (8) as: By a similar argument as above, we deduce that: Again by the Gauss equation for the Levi-Civita connection, we find that: Inserting (26) into (25), we have: By subtracting (24) from (27), we can state the following result: be an m-dimensional statistical warped product submanifold immersed into an n-dimensional statistical manifold of constant sectional curvaturec. Then: Further, we have: {h a 11 h a r+1,r+1 + h * a 11 h * a r+1,r+1 }, or we write it as: We use an optimization technique: For a ∈ [m + 1, n], we consider the quadratic forms: and: φ * a (h * a 11 , . . . , h * a The constrained extremum problem is max φ a subject to: Q : h a 11 + · · · + h a mm = t a , (t a is any constant).
As t a = ∑ m i=1 h a ii = (m − 1)α a , then we have: As φ a is obtained from the similar function studied in [39] by subtracting some square terms, φ a |Q will have the Hessian semi-negative definite. Consequently, the point in (31), together with (32) is a global maximum point, and hence, we calculate: Similarly, one gets: by considering (30) and the constrained extremum problem max φ * a subject to: Q * : h * a 11 + · · · + h * a mm = t * a , (t * a is any constant).
Thus, (28) becomes: By summarizing, we state the following: Theorem 4. Let N = N 1 × f N 2 be an m-dimensional statistical warped product submanifold immersed into an n-dimensional statistical manifold of constant sectional curvaturec. Then: By using (20), we obtain: For b = 1, 2, . . . , r, we also have: By summing up b from one to r, we find that: where ∆ N 1 and ∆ N 1 * are dual Laplacians of N 1 and ∆ N 1 0 denotes the Laplacian operator of N 1 for the Levi-Civita connection [37]. Thus, we have: be an m-dimensional statistical warped product submanifold immersed into an n-dimensional statistical manifold of constant sectional curvaturec. Then, the scalar curvature σ ∇,∇ * of N satisfies:

Optimal Casorati Inequality
Let {e 1 , . . . , e m } and {e m+1 , . . . , e n } be respectively the orthonormal basis of T p N and T ⊥ p N, p ∈ N. Then, the squared norm of second fundamental forms h and h * is denoted by C and C * , respectively, called the Casorati curvatures of N inÑ. Therefore, we have: where: If W is a q-dimensional subspace of TN, q ≥ 2, and {e 1 , . . . , e q } an orthonormal basis of W. Then, the scalar curvature of the q-plane section W is: S(e i , e j , e j , e i ), and the Casorati curvatures of the subspace W are as follows: (1) The normalized Casorati curvatures δ C (m − 1) and δ * C (m − 1) are defined as: )inf{C(W)|W : a hyperplane of T p N}, )inf{C * (W)|W : a hyperplane of T p N}.
Let ϕ : N = N 1 × f N 2 →Ñ(c) be an isometric statistical immersion of a warped product N 1 × f N 2 into a statistical manifold of constant sectional curvaturec. If we chose a local orthonormal frame {e 1 , . . . , e m } such that {e 1 , . . . , e r } are tangent to N 1 and {e r+1 , . . . , e r+k = e m } are tangent to N 2 , then the two partial mean curvature vectors H 1 (resp. H * 1 ) and H 2 (resp. H * 2 ) of N are given by: and: h(e r+j , e r+j ), h * (e r+j , e r+j ).
Furthermore, the Casorati curvatures are: and: Equation (21) implies: By using (8), the previous equation becomes: We define a polynomial P in terms of the components of the second fundamental form h 0 (with respect to the Levi-Civita connection) of N.
From (38), we find that the critical points of Q are the solutions of the following system of linear homogeneous equations.
for i ∈ {1, 2, . . . , r − 1} and a ∈ {m + 1, . . . , n}. Hence, every solution h 0c has: for i ∈ {1, 2, . . . , r − 1} and a ∈ {m + 1, . . . , n}. Now, we fix x ∈ Q. The bilinear form Θ : T x Q × T x Q → R has the following expression (cf. Theorem 1): where h denotes the second fundamental form of Q in R r and < ·, · > denotes the standard inner product on R r . The Hessian matrix of φ a is given by: Take a vector E ∈ T x Q, which satisfies a relation ∑ r i=1 E i = 0. As the hyperplane is totally geodesic, i.e., h = 0 in R r , we get: Θ(E, E) = Hess φ a (E, E) However, the point h 0c is the only optimal solution, i.e., the global minimum point of problem, and reaches a minimum Q(h 0c ) = 0 by considering (39) and the constrained extremum problem min ϕ a subject to: Q : h 0a 11 + · · · + h 0a kk = α a , (α a is any constant).

Example 2.
(R × z R n ,g = dt 2 + t 2 (dx 2 1 + · · · + dx 2 n ), ∇, ∇ * ) is a statistical warped product manifold, and it is isometric to the Euclidean (n + 1)-space E n+1 . Let N be a minimal submanifold of the unit hypersphere S n (1) ⊂ E n+1 center at the origin o ∈ E n+1 , and let C(N) be the cone over N with the vertex at o.
The metric of C(N) is the warped product metric g C(N) = dt 2 + t 2 g N , where g N denotes the metric of N. Any open submanifold M of C(N) is a warped product manifold, which admits an isometric minimal immersion into the statistical manifold E n+1 of constant sectional curvature zero.